## Year 2017-2018

Time: Tuesdays 3-4pm
Location: MW 154

## Schedule of talks:

 TIME SPEAKER TITLE August 29   Tue, 3pm Dave Anderson  (OSU) Determinantal formulas for degeneracy loci and Brill-Noether theory September 5   Tue, 3pm Hsian-Hua Tseng  (OSU) On the quantum K-rings of flag varieties of type A September 12   Tue, 3pm Angelica Cueto  (OSU) Tropical geometry of genus 2 curves September 19   Tue, 3pm Herb Clemens  (OSU) Eliptically-fibered Calabi-Yau (CY) threefolds and fourfolds in string theory October 3   Tue, 3pm Seth Baldwin  (UNC) Positivity in T-equivariant K-theory of flag varieties associated to Kac-Moody groups October 10   Tue, 3pm TBA  () TBA October 17   Tue, 3pm Roi Docampo  (Oklahoma) TBA October 24   Tue, 3pm TBA  () TBA October 26   Thu, 4:10pm CH 240 Renzo Cavalieri  (Colorado State U.) Colloquium talk October 31   Tue, 3pm Christin Bibby  (U. Michigan) TBA November 7   Tue, 3pm Chris Manon  (U. Kentucky) TBA November 14   Tue, 3pm Daniel Litt  (Columbia U.) TBA November 21   Tue, 3pm TBA  () TBA November 28   Tue, 3pm Jose Gonzalez  (UC Riverside) TBA December 5   Tue, 3pm Matthew Satriano  (U. Waterloo) TBA January 9   Tue, 3pm TBA  ( ) TBA January 16   Tue, 3pm TBA  () TBA January 23   Tue, 3pm William Fulton*  (Michigan) (*to be confirmed) January 30   Tue, 3pm TBA  () TBA February 6   Tue, 3pm TBA  () TBA February 13   Tue, 3pm TBA  () TBA February 20   Tue, 3pm TBA  () TBA February 27   Tue, 3pm TBA  () TBA March 6   Tue, 3pm TBA  () TBA March 13   Tue, 3pm (break) March 20   Tue, 3pm TBA  () TBA March 27   Tue, 3pm TBA  () TBA April 3   Tue, 3pm TBA  () TBA April 10   Tue, 3pm TBA  () TBA April 16-18   Tue, 4:10pm CH 240 Sergey Fomin  (U. Michigan) Zassenhaus Lecture Series 2018 April 24   Tue, 3pm TBA  () TBA

## Abstracts

(Anderson): Relations between degeneracy loci and divisors on curves go back over a hundred years: Giambelli's formula for the class of the locus where a matrix drops rank is a cornerstone of Schubert calculus; Brill and Noether's estimate for the dimension of the space of special divisors on a curve also comes from considerations of degeneracy loci. In this talk, I will describe work with Linda Chen and Nicola Tarasca which connects more recent developments in Schubert calculus with formulas by Eisenbud-Harris, Pirola, and Chan-Lopez-Pflueger-Teixidor for the genus of a Brill-Noether curve.

(Tseng): The quantum K-ring of a smooth projective variety X, introduced by Givental and YP Lee, is a deformation of the Grothendieck ring of coherent sheaves on X defined using holomorphic Euler characteristics on moduli spaces of stable maps to X (these are K-theoretic version of Gromov-Witten invariants). In this talk we discuss some properties of the quantum K-rings of X=Flr+1, the variety of complete flags in Cr+1, including:

(1) a presentation of the quantum K-ring;

(2) finiteness of quantum product;

(3) canonical polynomial representatives of Schubert classes.

This is based on joint work in progress with David Anderson and Linda Chen.

(Cueto): In this talk, I will discuss the structure of tropical and non-Archimedean analytic genus 2 curves and their moduli from three perspectives:

(1) as 2-to-1 covers of P1 branched at 6 points;

(2) as solutions to the hyperelliptic equation y2=f(x), where f has degree 5; and

(3) as metric graphs dual to genus 2 nodal algebraic curves over a valued field.

Our first description allows us to give an explicit combinatorial rule to characterize each such metric graph together with a harmonic map to a metric tree on 6 leaves, in terms of the valuations of 6 branch points in P1. Even though the tropicalization of a plane hyperelliptic curve shows no genus, we provide explicit re-embeddings of the input planar hyperelliptic curve that reveals the correct metric graphs.

From our third viewpoint, we consider the moduli space of abstract genus two tropical curves and translate the classical Igusa invariants characterizing isomorphism classes of genus two algebraic curves into the tropical realm. While these tropical Igusa functions do not yield coordinates on the tropical moduli space, we propose an alternative set of invariants that provides new length data. This is joint work with Hannah Markwig.

(Clemens): The talk will construct one example of the equivalence' between an elliptically-fibered CY-threefold endowed with a semi-stable E8 + E8 bundle and a semi-stable degeneration of an elliptically-fibered CY-fourfold. I will then loosely describe how this equivalence is used to break' the SU(5)-symmetry of grand unified theory (GUT) to the SU(3)xSU(2)xU(1)-symmetry of what physicists call the Standard Model of the physics of our cosmos.

(Baldwin): The cohomology ring of flag varieties has long been known to exhibit positivity properties. One such property is that the structure constants of the Schubert basis with respect to the cup product are non-negative. Brion (2002) and Anderson-Griffeth-Miller (2011) have shown that positivity extends to K-theory and T-equivariant K-theory, respectively. In this talk I will discuss recent work (joint with Shrawan Kumar) which generalizes these results to the case of Kac-Moody groups.

## Past Seminars

Ohio State University Algebraic Geometry Seminar-Year 2016-2017

Ohio State University Algebraic Geometry Seminar-Year 2015-2016

Ohio State University Algebraic Geometry Seminar-Year 2014-2015

Ohio State University Algebraic Geometry Seminar-Year 2013-2014

This page is maintained by Angie Cueto and Dave Anderson.